Applying Corollary REF  with \(r=1\) , \(\theta = \min \lbrace r,p\rbrace \) , taking into account Remarks REF , REF , we obtain in view of Theorem 2.5 of [1]}
that, for \(p,q \in (0,\infty ]\) , \(\lbrace s_{k}\rbrace  \in Y^{\alpha _{3}}_{\alpha _{1},\alpha _{2}}\) , \(\alpha _{1} > n\max \lbrace \frac{1}{p}-1,0\rbrace \) , \(l > \alpha _{2}\) ,
the space \(B^{\lbrace s_{k}\rbrace }_{p,q}(\mathbb {R}^{n})\)  consists of the functions \(f \in L^{\text{\rm loc}}_{\max \lbrace 1,p\rbrace }(\mathbb {R}^{n})\)  (moreover,
\(B^{\lbrace s_{k}\rbrace }_{p,q}(\mathbb {R}^{n})\)  is continuously embedded into \(L^{\text{\rm loc}}_{\max \lbrace 1,p\rbrace }(\mathbb {R}^{n})\) ). Besides,
\(\Vert f|B^{\lbrace s_{k}\rbrace }_{p,q}(\mathbb {R}^{n})\Vert  \sim \left(\sum \limits _{k=1}^{\infty }\Vert s_{k}\overline{\Delta }^{l}_{1}(2^{-k})f|L_{p}(\mathbb {R}^{n})\Vert ^{q}\right)^{\frac{1}{q}}+\Vert s_{0}\Vert f|L_{1}(\cdot +I^{n})\Vert |L_{p}(\mathbb {R}^{n})\Vert \) 
